In traditional centralized machine learning, transmitting raw data to a cloud center incurs high communication costs and raises privacy concerns. This is particularly challenging in mobile edge environments, where devices are dynamic and resource-constrained. Federated Learning (FL) addresses these issues by allowing devices to train models locally and upload parameters to a central server without sharing local data. However, limited wireless channel resources and dynamic transmission performance make communication overhead a major bottleneck of FL in mobile edge environments. Concerning this challenge, this survey provides a comprehensive summary of methods to improve communication efficiency in FL, focusing on: 1) minimizing communication complexity to reduce total transmission volume, 2) scheduling resources appropriately to improve training efficiency, 3) utilizing over-the-air computation (OTA) to integrate computation into communication for accommodating the computation/communication characteristics of FL in mobile edge environments. Thus, this work analyzes research from the perspective of convergence and data heterogeneity to reduce communication rounds by optimizing algorithm performance. We hope that this survey could offer insights into communication-efficient FL for future research.

Federated Learning (FL) has emerged as a promising technique for collaboratively training machine learning models among multiple participants while preserving privacy-sensitive data. However, the conventional parameter server architecture presents challenges in terms of communication overhead when employing iterative optimization methods such as Stochastic Gradient Descent (SGD). Although communication compression techniques can reduce the traffic cost of FL during each training round, they often lead to degraded convergence rates, mainly due to compression errors and data heterogeneity. To address these issues, this paper presents FedQClip, an innovative approach that combines quantization and Clipped SGD. FedQClip leverages an adaptive step size inversely proportional to the $\ell_{2}$ norm of the gradient, effectively mitigating the negative impacts of quantized errors. Additionally, clipped operations can be applied locally and globally to further expedite training. Theoretical analyses provide evidence that, even under the settings of Non-IID (non-independent and identically distributed) data, FedQClip achieves a convergence rate of $\mathcal{O}(\frac{1}{\sqrt{T}})$, effectively addressing the convergence degradation caused by compression errors. Furthermore, our theoretical analysis highlights the importance of selecting an appropriate number of local updates to enhance the convergence of FL training. Through extensive experiments, we demonstrate that FedQClip outperforms state-of-the-art methods in terms of communication efficiency and convergence rate.